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■❈❚❆❈ ✷✵✶✻ ❲❡❞♥❡s❞❛② ✷✻t❤ ♦❢ ❖❝t♦❜❡r✱ ✷✵✶✻ 臺北，臺灣

P❛r❛♠❡tr✐❝ ❉❡❛❞❧♦❝❦✲❋r❡❡♥❡ss ❈❤❡❝❦✐♥❣ ❚✐♠❡❞ ❆✉t♦♠❛t❛

➱t✐❡♥♥❡ ❆♥❞ré1,2

1 ▲■P◆✱ ❯♥✐✈❡rs✐té P❛r✐s ✶✸✱ ❙♦r❜♦♥♥❡ P❛r✐s ❈✐té✱ ❈◆❘❙✱ ❋r❛♥❝❡ 2 ➱❝♦❧❡ ❈❡♥tr❛❧❡ ❞❡ ◆❛♥t❡s✱ ■❘❈❈②◆✱ ❈◆❘❙✱ ❯▼❘ ✻✺✾✼✱ ❋r❛♥❝❡ ➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶ ✴ ✷✷

■♥tr♦❞✉❝t✐♦♥

❈♦♥t❡①t✿ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

❚✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

y = ❞❡❧❛② x := 0 x < ♣❡r✐♦❞

❆ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ ✐s ✉♥r❡❛❝❤❛❜❧❡ ❆ ♣r♦♣❡rt② t♦ ❜❡ s❛t✐s✜❡❞ ◗✉❡st✐♦♥✿ ❨❡s ✐❢✳ ✳ ✳ ◆♦

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✷ ✴ ✷✷

■♥tr♦❞✉❝t✐♦♥

❈♦♥t❡①t✿ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

❚✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

y = ❞❡❧❛② x := 0 x < ♣❡r✐♦❞

❆ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ ?

| =

✐s ✉♥r❡❛❝❤❛❜❧❡ ❆ ♣r♦♣❡rt② t♦ ❜❡ s❛t✐s✜❡❞ ◗✉❡st✐♦♥✿ ❞♦❡s t❤❡ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ s❛t✐s❢② t❤❡ ♣r♦♣❡rt②❄ ❨❡s ✐❢✳ ✳ ✳ ◆♦

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✷ ✴ ✷✷

■♥tr♦❞✉❝t✐♦♥

❈♦♥t❡①t✿ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

❚✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

y = ❞❡❧❛② x := 0 x < ♣❡r✐♦❞

❆ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ ?

| =

✐s ✉♥r❡❛❝❤❛❜❧❡ ❆ ♣r♦♣❡rt② t♦ ❜❡ s❛t✐s✜❡❞ ◗✉❡st✐♦♥✿ ❞♦❡s t❤❡ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ s❛t✐s❢② t❤❡ ♣r♦♣❡rt②❄ ❨❡s ✐❢✳ ✳ ✳ ◆♦ ❈♦✉♥t❡r❡①❛♠♣❧❡

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✷ ✴ ✷✷

■♥tr♦❞✉❝t✐♦♥

❈♦♥t❡①t✿ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

❚✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

y = ❞❡❧❛② x := 0 x < ♣❡r✐♦❞

❆ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ ?

| =

✐s ✉♥r❡❛❝❤❛❜❧❡ ❆ ♣r♦♣❡rt② t♦ ❜❡ s❛t✐s✜❡❞ ◗✉❡st✐♦♥✿ ❢♦r ✇❤❛t ✈❛❧✉❡s ♦❢ t❤❡ ♣❛r❛♠❡t❡rs ❞♦❡s t❤❡ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ s❛t✐s❢② t❤❡ ♣r♦♣❡rt②❄ ❨❡s ✐❢✳ ✳ ✳ ◆♦ 2❞❡❧❛② > ♣❡r✐♦❞ ∧ ♣❡r✐♦❞ < 20.46

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✷ ✴ ✷✷

❖✉t❧✐♥❡

❖✉t❧✐♥❡

✶

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣

✸

❇❛❝❦✇❛r❞ ❯♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥

✹

❊①♣❡r✐♠❡♥ts

✺

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✸ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❖✉t❧✐♥❡

✶

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣

✸

❇❛❝❦✇❛r❞ ❯♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥

✹

❊①♣❡r✐♠❡♥ts

✺

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✹ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭❚❆✮

❋✐♥✐t❡ st❛t❡ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s✮ ❛♥❞ ❛❝t✐♦♥s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t ♦❢ ❝❧♦❝❦s ❬❆❧✉r ❛♥❞ ❉✐❧❧✱ ✶✾✾✹✱ ❍❡♥③✐♥❣❡r ❡t ❛❧✳✱ ✶✾✾✹❪

❘❡❛❧✲✈❛❧✉❡❞ ✈❛r✐❛❜❧❡s ❡✈♦❧✈✐♥❣ ❧✐♥❡❛r❧② ❛t t❤❡ s❛♠❡ r❛t❡

❋❡❛t✉r❡s

▲♦❝❛t✐♦♥ ✐♥✈❛r✐❛♥t✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ st❛② ❛t ❛ ❧♦❝❛t✐♦♥ ❚r❛♥s✐t✐♦♥ ❣✉❛r❞✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ ❡♥❛❜❧❡ ❛ tr❛♥s✐t✐♦♥ ❈❧♦❝❦ r❡s❡t✿ s♦♠❡ ♦❢ t❤❡ ❝❧♦❝❦s ❝❛♥ ❜❡ s❡t t♦ ❛t ❡❛❝❤ tr❛♥s✐t✐♦♥

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✺ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭❚❆✮

❋✐♥✐t❡ st❛t❡ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞ ❛❝t✐♦♥s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t ♦❢ ❝❧♦❝❦s ❬❆❧✉r ❛♥❞ ❉✐❧❧✱ ✶✾✾✹✱ ❍❡♥③✐♥❣❡r ❡t ❛❧✳✱ ✶✾✾✹❪

❘❡❛❧✲✈❛❧✉❡❞ ✈❛r✐❛❜❧❡s ❡✈♦❧✈✐♥❣ ❧✐♥❡❛r❧② ❛t t❤❡ s❛♠❡ r❛t❡

❋❡❛t✉r❡s

▲♦❝❛t✐♦♥ ✐♥✈❛r✐❛♥t✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ st❛② ❛t ❛ ❧♦❝❛t✐♦♥ ❚r❛♥s✐t✐♦♥ ❣✉❛r❞✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ ❡♥❛❜❧❡ ❛ tr❛♥s✐t✐♦♥ ❈❧♦❝❦ r❡s❡t✿ s♦♠❡ ♦❢ t❤❡ ❝❧♦❝❦s ❝❛♥ ❜❡ s❡t t♦ ❛t ❡❛❝❤ tr❛♥s✐t✐♦♥

♣r❡ss❄ ❝✉♣✦ ♣r❡ss❄ ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✺ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭❚❆✮

❋✐♥✐t❡ st❛t❡ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞ ❛❝t✐♦♥s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t X ♦❢ ❝❧♦❝❦s ❬❆❧✉r ❛♥❞ ❉✐❧❧✱ ✶✾✾✹✱ ❍❡♥③✐♥❣❡r ❡t ❛❧✳✱ ✶✾✾✹❪

❘❡❛❧✲✈❛❧✉❡❞ ✈❛r✐❛❜❧❡s ❡✈♦❧✈✐♥❣ ❧✐♥❡❛r❧② ❛t t❤❡ s❛♠❡ r❛t❡

❋❡❛t✉r❡s

▲♦❝❛t✐♦♥ ✐♥✈❛r✐❛♥t✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ st❛② ❛t ❛ ❧♦❝❛t✐♦♥ ❚r❛♥s✐t✐♦♥ ❣✉❛r❞✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ ❡♥❛❜❧❡ ❛ tr❛♥s✐t✐♦♥ ❈❧♦❝❦ r❡s❡t✿ s♦♠❡ ♦❢ t❤❡ ❝❧♦❝❦s ❝❛♥ ❜❡ s❡t t♦ ❛t ❡❛❝❤ tr❛♥s✐t✐♦♥

♣r❡ss❄ ❝✉♣✦ ♣r❡ss❄ ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✺ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭❚❆✮

❋✐♥✐t❡ st❛t❡ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞ ❛❝t✐♦♥s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t X ♦❢ ❝❧♦❝❦s ❬❆❧✉r ❛♥❞ ❉✐❧❧✱ ✶✾✾✹✱ ❍❡♥③✐♥❣❡r ❡t ❛❧✳✱ ✶✾✾✹❪

❘❡❛❧✲✈❛❧✉❡❞ ✈❛r✐❛❜❧❡s ❡✈♦❧✈✐♥❣ ❧✐♥❡❛r❧② ❛t t❤❡ s❛♠❡ r❛t❡

❋❡❛t✉r❡s

▲♦❝❛t✐♦♥ ✐♥✈❛r✐❛♥t✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ st❛② ❛t ❛ ❧♦❝❛t✐♦♥ ❚r❛♥s✐t✐♦♥ ❣✉❛r❞✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ ❡♥❛❜❧❡ ❛ tr❛♥s✐t✐♦♥ ❈❧♦❝❦ r❡s❡t✿ s♦♠❡ ♦❢ t❤❡ ❝❧♦❝❦s ❝❛♥ ❜❡ s❡t t♦ ❛t ❡❛❝❤ tr❛♥s✐t✐♦♥

y ≤ 5 y ≤ 8 ♣r❡ss❄ ❝✉♣✦ ♣r❡ss❄ ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✺ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭❚❆✮

❋✐♥✐t❡ st❛t❡ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞ ❛❝t✐♦♥s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t X ♦❢ ❝❧♦❝❦s ❬❆❧✉r ❛♥❞ ❉✐❧❧✱ ✶✾✾✹✱ ❍❡♥③✐♥❣❡r ❡t ❛❧✳✱ ✶✾✾✹❪

❘❡❛❧✲✈❛❧✉❡❞ ✈❛r✐❛❜❧❡s ❡✈♦❧✈✐♥❣ ❧✐♥❡❛r❧② ❛t t❤❡ s❛♠❡ r❛t❡

❋❡❛t✉r❡s

▲♦❝❛t✐♦♥ ✐♥✈❛r✐❛♥t✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ st❛② ❛t ❛ ❧♦❝❛t✐♦♥ ❚r❛♥s✐t✐♦♥ ❣✉❛r❞✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ ❡♥❛❜❧❡ ❛ tr❛♥s✐t✐♦♥ ❈❧♦❝❦ r❡s❡t✿ s♦♠❡ ♦❢ t❤❡ ❝❧♦❝❦s ❝❛♥ ❜❡ s❡t t♦ ❛t ❡❛❝❤ tr❛♥s✐t✐♦♥

y ≤ 5 y ≤ 8 ♣r❡ss❄ y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ y = 8 ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✺ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭❚❆✮

❋✐♥✐t❡ st❛t❡ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞ ❛❝t✐♦♥s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t X ♦❢ ❝❧♦❝❦s ❬❆❧✉r ❛♥❞ ❉✐❧❧✱ ✶✾✾✹✱ ❍❡♥③✐♥❣❡r ❡t ❛❧✳✱ ✶✾✾✹❪

❘❡❛❧✲✈❛❧✉❡❞ ✈❛r✐❛❜❧❡s ❡✈♦❧✈✐♥❣ ❧✐♥❡❛r❧② ❛t t❤❡ s❛♠❡ r❛t❡

❋❡❛t✉r❡s

▲♦❝❛t✐♦♥ ✐♥✈❛r✐❛♥t✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ st❛② ❛t ❛ ❧♦❝❛t✐♦♥ ❚r❛♥s✐t✐♦♥ ❣✉❛r❞✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ ❡♥❛❜❧❡ ❛ tr❛♥s✐t✐♦♥ ❈❧♦❝❦ r❡s❡t✿ s♦♠❡ ♦❢ t❤❡ ❝❧♦❝❦s ❝❛♥ ❜❡ s❡t t♦ 0 ❛t ❡❛❝❤ tr❛♥s✐t✐♦♥

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✺ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❈♦♥❝r❡t❡ s❡♠❛♥t✐❝s ♦❢ t✐♠❡❞ ❛✉t♦♠❛t❛

❈♦♥❝r❡t❡ st❛t❡ ♦❢ ❛ ❚❆✿ ♣❛✐r (l, w)✱ ✇❤❡r❡

l ✐s ❛ ❧♦❝❛t✐♦♥✱ w ✐s ❛ ✈❛❧✉❛t✐♦♥ ♦❢ ❡❛❝❤ ❝❧♦❝❦

❈♦♥❝r❡t❡ r✉♥✿ ❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❝♦♥❝r❡t❡ st❛t❡s ❛♥❞ ❛❝t✐♦♥s ♦r t✐♠❡ ❡❧❛♣s❡

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✻ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r ❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 15.4

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 15.4 ♣r❡ss❄

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 15.4 ♣r❡ss❄ 5

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 15.4 ♣r❡ss❄ 5 ❝✉♣✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y ♣r❡ss❄

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 ♣r❡ss❄ 1.5

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 ♣r❡ss❄ 1.5 ♣r❡ss❄

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 2.7 4.2 ♣r❡ss❄ 1.5 ♣r❡ss❄ 2.7

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 2.7 4.2 4.2 ♣r❡ss❄ 1.5 ♣r❡ss❄ 2.7 ♣r❡ss❄

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 2.7 4.2 4.2 0.8 5 ♣r❡ss❄ 1.5 ♣r❡ss❄ 2.7 ♣r❡ss❄ 0.8

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 2.7 4.2 4.2 0.8 5 0.8 5 ♣r❡ss❄ 1.5 ♣r❡ss❄ 2.7 ♣r❡ss❄ 0.8 ❝✉♣✦

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 2.7 4.2 4.2 0.8 5 0.8 5 3.8 8 ♣r❡ss❄ 1.5 ♣r❡ss❄ 2.7 ♣r❡ss❄ 0.8 ❝✉♣✦ 3

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 2.7 4.2 4.2 0.8 5 0.8 5 3.8 8 3.8 8 ♣r❡ss❄ 1.5 ♣r❡ss❄ 2.7 ♣r❡ss❄ 0.8 ❝✉♣✦ 3 ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✼ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

P❛r❛♠❡tr✐❝ t✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭P❚❆✮

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s✱ ❛❝t✐♦♥s ❛♥❞ ❝❧♦❝❦s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t ♦❢ ♣❛r❛♠❡t❡rs

❬❆❧✉r ❡t ❛❧✳✱ ✶✾✾✸❪

❯♥❦♥♦✇♥ ❝♦♥st❛♥ts ✉s❡❞ ✐♥ ❣✉❛r❞s ❛♥❞ ✐♥✈❛r✐❛♥ts

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✽ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

P❛r❛♠❡tr✐❝ t✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭P❚❆✮

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s✱ ❛❝t✐♦♥s ❛♥❞ ❝❧♦❝❦s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t P ♦❢ ♣❛r❛♠❡t❡rs

❬❆❧✉r ❡t ❛❧✳✱ ✶✾✾✸❪

❯♥❦♥♦✇♥ ❝♦♥st❛♥ts ✉s❡❞ ✐♥ ❣✉❛r❞s ❛♥❞ ✐♥✈❛r✐❛♥ts

y ≤ p2 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = p2 ❝✉♣✦ x ≥ p1 ♣r❡ss❄ x := 0 y = p3 ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✽ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❱❛❧✉❛t✐♦♥ ♦❢ ❛ P❚❆

• ✐✈❡♥ ❛ P❚❆ A ❛♥❞ ❛ ♣❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥ v✱ ✇❡ ❞❡♥♦t❡ ❜② v(A)

t❤❡ ✭♥♦♥✲♣❛r❛♠❡tr✐❝✮ t✐♠❡❞ ❛✉t♦♠❛t♦♥ ✇❤❡r❡ ❛❧❧ ♣❛r❛♠❡t❡rs ❛r❡ ✈❛❧✉❛t❡❞ ❜② v

♣r❡ss❄ ❝✉♣✦ ♣r❡ss❄ ❝♦✛❡❡✦ ♣r❡ss❄ ❝✉♣✦ ♣r❡ss❄ ❝♦✛❡❡✦

✇✐t❤

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✾ ✴ ✷✷

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❱❛❧✉❛t✐♦♥ ♦❢ ❛ P❚❆

• ✐✈❡♥ ❛ P❚❆ A ❛♥❞ ❛ ♣❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥ v✱ ✇❡ ❞❡♥♦t❡ ❜② v(A)

t❤❡ ✭♥♦♥✲♣❛r❛♠❡tr✐❝✮ t✐♠❡❞ ❛✉t♦♠❛t♦♥ ✇❤❡r❡ ❛❧❧ ♣❛r❛♠❡t❡rs ❛r❡ ✈❛❧✉❛t❡❞ ❜② v

v       

y ≤ p2 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = p2 ❝✉♣✦ x ≥ p1 ♣r❡ss❄ x := 0 y = p3 ❝♦✛❡❡✦

       =

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

✇✐t❤ v :    p1 → 1 p2 → 5 p3 → 8

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✾ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣

❖✉t❧✐♥❡

✶

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣

✸

❇❛❝❦✇❛r❞ ❯♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥

✹

❊①♣❡r✐♠❡♥ts

✺

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✵ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ Pr♦❜❧❡♠

❖❜❥❡❝t✐✈❡

Pr♦❜❧❡♠

• ✐✈❡♥ ❛ P❚❆ A✱ s②♥t❤❡s✐③❡ ✈❛❧✉❛t✐♦♥s v ❢♦r ✇❤✐❝❤ v(A) ✐s

❞❡❛❞❧♦❝❦✲❢r❡❡✳ ❉❡❛❞❧♦❝❦✿ st❛t❡ ✐♥ ✇❤✐❝❤ ♥♦ ❞✐s❝r❡t❡ ❛❝t✐♦♥ ♠❛② ❜❡ t❛❦❡♥ ✐♥ s♦♠❡ st❛t❡✱ ❡✈❡♥ ❛❢t❡r ❡❧❛♣s✐♥❣ s♦♠❡ t✐♠❡✳ ❚❤❡ ♠❡r❡ ❡♠♣t✐♥❡ss ♦❢ t❤✐s ✈❛❧✉❛t✐♦♥ s❡t ✐s ✉♥❞❡❝✐❞❛❜❧❡ ❢♦r P❚❆s✱ ❛♥❞ ❡✈❡♥ ❢♦r s✉❜❝❧❛ss❡s ❦♥♦✇♥ ❢♦r ♦t❤❡r ❞❡❝✐❞❛❜✐❧✐t② r❡s✉❧ts

❬❆♥❞ré ❛♥❞ ▲✐♠❡✱ ✷✵✶✻❪

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✶ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ Pr♦❜❧❡♠

❖❜❥❡❝t✐✈❡

Pr♦❜❧❡♠

• ✐✈❡♥ ❛ P❚❆ A✱ s②♥t❤❡s✐③❡ ✈❛❧✉❛t✐♦♥s v ❢♦r ✇❤✐❝❤ v(A) ✐s

❞❡❛❞❧♦❝❦✲❢r❡❡✳ ❉❡❛❞❧♦❝❦✿ st❛t❡ ✐♥ ✇❤✐❝❤ ♥♦ ❞✐s❝r❡t❡ ❛❝t✐♦♥ ♠❛② ❜❡ t❛❦❡♥ ✐♥ s♦♠❡ st❛t❡✱ ❡✈❡♥ ❛❢t❡r ❡❧❛♣s✐♥❣ s♦♠❡ t✐♠❡✳ ❚❤❡ ♠❡r❡ ❡♠♣t✐♥❡ss ♦❢ t❤✐s ✈❛❧✉❛t✐♦♥ s❡t ✐s ✉♥❞❡❝✐❞❛❜❧❡ ❢♦r P❚❆s✱ ❛♥❞ ❡✈❡♥ ❢♦r s✉❜❝❧❛ss❡s ❦♥♦✇♥ ❢♦r ♦t❤❡r ❞❡❝✐❞❛❜✐❧✐t② r❡s✉❧ts

❬❆♥❞ré ❛♥❞ ▲✐♠❡✱ ✷✵✶✻❪

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✶ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ Pr♦❜❧❡♠

❖❜❥❡❝t✐✈❡

Pr♦❜❧❡♠

• ✐✈❡♥ ❛ P❚❆ A✱ s②♥t❤❡s✐③❡ ✈❛❧✉❛t✐♦♥s v ❢♦r ✇❤✐❝❤ v(A) ✐s

❞❡❛❞❧♦❝❦✲❢r❡❡✳ ❉❡❛❞❧♦❝❦✿ st❛t❡ ✐♥ ✇❤✐❝❤ ♥♦ ❞✐s❝r❡t❡ ❛❝t✐♦♥ ♠❛② ❜❡ t❛❦❡♥ ✐♥ s♦♠❡ st❛t❡✱ ❡✈❡♥ ❛❢t❡r ❡❧❛♣s✐♥❣ s♦♠❡ t✐♠❡✳ ❚❤❡ ♠❡r❡ ❡♠♣t✐♥❡ss ♦❢ t❤✐s ✈❛❧✉❛t✐♦♥ s❡t ✐s ✉♥❞❡❝✐❞❛❜❧❡ ❢♦r P❚❆s✱ ❛♥❞ ❡✈❡♥ ❢♦r s✉❜❝❧❛ss❡s ❦♥♦✇♥ ❢♦r ♦t❤❡r ❞❡❝✐❞❛❜✐❧✐t② r❡s✉❧ts

❬❆♥❞ré ❛♥❞ ▲✐♠❡✱ ✷✵✶✻❪

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✶ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ Pr♦❜❧❡♠

❊①❛♠♣❧❡s

① ≤ p1 + 5 ① ≤ 10 ① ≥ p2 ❛ ❛ ① := 0

❉❡❛❞❧♦❝❦✲❢r❡❡ ✐❢

① ≤ p ❛ ❛

❉❡❛❞❧♦❝❦✲❢r❡❡ ❢♦r ♥♦ ✈❛❧✉❛t✐♦♥

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✷ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ Pr♦❜❧❡♠

❊①❛♠♣❧❡s

① ≤ p1 + 5 ① ≤ 10 ① ≥ p2 ❛ ❛ ① := 0

❉❡❛❞❧♦❝❦✲❢r❡❡ ✐❢ p1 + 5 ≥ p2 ≤ 10

① ≤ p ❛ ❛

❉❡❛❞❧♦❝❦✲❢r❡❡ ❢♦r ♥♦ ✈❛❧✉❛t✐♦♥

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✷ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ Pr♦❜❧❡♠

❊①❛♠♣❧❡s

① ≤ p1 + 5 ① ≤ 10 ① ≥ p2 ❛ ❛ ① := 0

❉❡❛❞❧♦❝❦✲❢r❡❡ ✐❢ p1 + 5 ≥ p2 ≤ 10

① ≤ p ❛ ❛

❉❡❛❞❧♦❝❦✲❢r❡❡ ❢♦r ♥♦ ✈❛❧✉❛t✐♦♥

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✷ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ ❙❡♠❛♥t✐❝s

❙②♠❜♦❧✐❝ s❡♠❛♥t✐❝s ♦❢ P❚❆s

❙②♠❜♦❧✐❝ st❛t❡ s = (l, Z)✿ ❧♦❝❛t✐♦♥ ✰ ❝♦♥✈❡① ♣♦❧②❤❡❞r♦♥ ❝♦♥str❛✐♥✐♥❣ ❜♦t❤ ❝❧♦❝❦s ❛♥❞ ♣❛r❛♠❡t❡rs

❬❆♥❞ré ❡t ❛❧✳✱ ✷✵✵✾✱ ❏♦✈❛♥♦✈✐➣ ❡t ❛❧✳✱ ✷✵✶✺❪

❈♦♥✈❡① ♣♦❧②❤❡❞r❛ ♦❜t❛✐♥❡❞ ❤❛✈❡ ❛ s♣❡❝✐❛❧ ❢♦r♠ ❝❛❧❧❡❞ ♣❛r❛♠❡tr✐❝ ③♦♥❡

❬❍✉♥❡ ❡t ❛❧✳✱ ✷✵✵✷❪

y ≤ p x ≥ p2 y := 0 Z0 =    x = y ∧ 0 ≤ y ≤ p1 ∧ p1, p2 ≥ 0 Z1 =    p2 ≤ x − y ≤ p1 ∧ (p2 ≤ p1) ∧ x, y, p1, p2 ≥ 0 ◆♦t❡✿ t❤❡r❡ ✐s ❛ ♣♦t❡♥t✐❛❧❧② ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ s②♠❜♦❧✐❝ st❛t❡s

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✸ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ ❙❡♠❛♥t✐❝s

❙②♠❜♦❧✐❝ s❡♠❛♥t✐❝s ♦❢ P❚❆s

❙②♠❜♦❧✐❝ st❛t❡ s = (l, Z)✿ ❧♦❝❛t✐♦♥ ✰ ❝♦♥✈❡① ♣♦❧②❤❡❞r♦♥ ❝♦♥str❛✐♥✐♥❣ ❜♦t❤ ❝❧♦❝❦s ❛♥❞ ♣❛r❛♠❡t❡rs

❬❆♥❞ré ❡t ❛❧✳✱ ✷✵✵✾✱ ❏♦✈❛♥♦✈✐➣ ❡t ❛❧✳✱ ✷✵✶✺❪

❈♦♥✈❡① ♣♦❧②❤❡❞r❛ ♦❜t❛✐♥❡❞ ❤❛✈❡ ❛ s♣❡❝✐❛❧ ❢♦r♠ ❝❛❧❧❡❞ ♣❛r❛♠❡tr✐❝ ③♦♥❡

❬❍✉♥❡ ❡t ❛❧✳✱ ✷✵✵✷❪

y ≤ p x ≥ p2 y := 0 Z0 =    x = y ∧ 0 ≤ y ≤ p1 ∧ p1, p2 ≥ 0 Z1 =    p2 ≤ x − y ≤ p1 ∧ (p2 ≤ p1) ∧ x, y, p1, p2 ≥ 0 ◆♦t❡✿ t❤❡r❡ ✐s ❛ ♣♦t❡♥t✐❛❧❧② ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ s②♠❜♦❧✐❝ st❛t❡s

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✸ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ ❆❧❣♦r✐t❤♠

❆ s❡♠✐✲❛❧❣♦r✐t❤♠ ❢♦r ❞❡❛❞❧♦❝❦✲❡①✐st❡♥❝❡

❉❙②♥t❤(s, P❛ss❡❞) =                                   

r❡t✉r♥ ♥♦t❤✐♥❣ ✐❢ st❛t❡ ♠❡t ❜❡❢♦r❡

✐❢ s P❛ss❡❞ ♦t❤❡r✇✐s❡✿

r❡❝✉rs✐✈❡ ❝❛❧❧ ♦♥ t❤❡ s✉❝❝❡ss♦rs s ❙✉❝❝ s

❉❙②♥t❤ s P❛ss❡❞ s s

s ❙✉❝❝ s ❝❛♥ ♣❛ss t❤❡ ❣✉❛r❞

s s s s

✈❛❧✉❛t✐♦♥s ❢♦r ✇❤✐❝❤ ♦♥❡ ♠❛② ❣❡t st✉❝❦ ✐♥ s

s s′ s′′

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✹ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ ❆❧❣♦r✐t❤♠

❆ s❡♠✐✲❛❧❣♦r✐t❤♠ ❢♦r ❞❡❛❞❧♦❝❦✲❡①✐st❡♥❝❡

❉❙②♥t❤(s, P❛ss❡❞) =                                   

r❡t✉r♥ ♥♦t❤✐♥❣ ✐❢ st❛t❡ ♠❡t ❜❡❢♦r❡

• ✐❢ s

P❛ss❡❞ ♦t❤❡r✇✐s❡✿

r❡❝✉rs✐✈❡ ❝❛❧❧ ♦♥ t❤❡ s✉❝❝❡ss♦rs s ❙✉❝❝ s

❉❙②♥t❤ s P❛ss❡❞ s s

s ❙✉❝❝ s ❝❛♥ ♣❛ss t❤❡ ❣✉❛r❞

s s s s

✈❛❧✉❛t✐♦♥s ❢♦r ✇❤✐❝❤ ♦♥❡ ♠❛② ❣❡t st✉❝❦ ✐♥ s

s s′ s′′

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✹ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ ❆❧❣♦r✐t❤♠

❆ s❡♠✐✲❛❧❣♦r✐t❤♠ ❢♦r ❞❡❛❞❧♦❝❦✲❡①✐st❡♥❝❡

❉❙②♥t❤(s, P❛ss❡❞) =                                   

r❡t✉r♥ ♥♦t❤✐♥❣ ✐❢ st❛t❡ ♠❡t ❜❡❢♦r❡

• ⊥ ✐❢ s ∈ P❛ss❡❞

♦t❤❡r✇✐s❡✿

r❡❝✉rs✐✈❡ ❝❛❧❧ ♦♥ t❤❡ s✉❝❝❡ss♦rs s ❙✉❝❝ s

❉❙②♥t❤ s P❛ss❡❞ s s

s ❙✉❝❝ s ❝❛♥ ♣❛ss t❤❡ ❣✉❛r❞

s s s s

✈❛❧✉❛t✐♦♥s ❢♦r ✇❤✐❝❤ ♦♥❡ ♠❛② ❣❡t st✉❝❦ ✐♥ s

s s′ s′′

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✹ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ ❆❧❣♦r✐t❤♠

❆ s❡♠✐✲❛❧❣♦r✐t❤♠ ❢♦r ❞❡❛❞❧♦❝❦✲❡①✐st❡♥❝❡

❉❙②♥t❤(s, P❛ss❡❞) =                                   

r❡t✉r♥ ♥♦t❤✐♥❣ ✐❢ st❛t❡ ♠❡t ❜❡❢♦r❡

• ⊥ ✐❢ s ∈ P❛ss❡❞

♦t❤❡r✇✐s❡✿

r❡❝✉rs✐✈❡ ❝❛❧❧ ♦♥ t❤❡ s✉❝❝❡ss♦rs

• s

❙✉❝❝ s

❉❙②♥t❤ s P❛ss❡❞ s s

s ❙✉❝❝ s ❝❛♥ ♣❛ss t❤❡ ❣✉❛r❞

s s s s

• ✈❛❧✉❛t✐♦♥s ❢♦r ✇❤✐❝❤ ♦♥❡ ♠❛② ❣❡t st✉❝❦ ✐♥ s

s s′ s′′

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✹ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ ❆❧❣♦r✐t❤♠

❆ s❡♠✐✲❛❧❣♦r✐t❤♠ ❢♦r ❞❡❛❞❧♦❝❦✲❡①✐st❡♥❝❡

❉❙②♥t❤(s, P❛ss❡❞) =                                   

r❡t✉r♥ ♥♦t❤✐♥❣ ✐❢ st❛t❡ ♠❡t ❜❡❢♦r❡

• ⊥ ✐❢ s ∈ P❛ss❡❞

♦t❤❡r✇✐s❡✿

r❡❝✉rs✐✈❡ ❝❛❧❧ ♦♥ t❤❡ s✉❝❝❡ss♦rs

• s′∈❙✉❝❝(s)

❉❙②♥t❤(s′, P❛ss❡❞ ∪ {s})

• s

s ❙✉❝❝ s ❝❛♥ ♣❛ss t❤❡ ❣✉❛r❞

s s s s

• ✈❛❧✉❛t✐♦♥s ❢♦r ✇❤✐❝❤ ♦♥❡ ♠❛② ❣❡t st✉❝❦ ✐♥ s

s s′ s′′

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✹ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ ❆❧❣♦r✐t❤♠

❆ s❡♠✐✲❛❧❣♦r✐t❤♠ ❢♦r ❞❡❛❞❧♦❝❦✲❡①✐st❡♥❝❡

❉❙②♥t❤(s, P❛ss❡❞) =                                   

r❡t✉r♥ ♥♦t❤✐♥❣ ✐❢ st❛t❡ ♠❡t ❜❡❢♦r❡

• ⊥ ✐❢ s ∈ P❛ss❡❞

♦t❤❡r✇✐s❡✿

r❡❝✉rs✐✈❡ ❝❛❧❧ ♦♥ t❤❡ s✉❝❝❡ss♦rs

• s′∈❙✉❝❝(s)

❉❙②♥t❤(s′, P❛ss❡❞ ∪ {s})

• ∪

s

s ❙✉❝❝ s ❝❛♥ ♣❛ss t❤❡ ❣✉❛r❞

s s s s

• ✈❛❧✉❛t✐♦♥s ❢♦r ✇❤✐❝❤ ♦♥❡ ♠❛② ❣❡t st✉❝❦ ✐♥ s

s s′ s′′

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✹ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ ❆❧❣♦r✐t❤♠

❆ s❡♠✐✲❛❧❣♦r✐t❤♠ ❢♦r ❞❡❛❞❧♦❝❦✲❡①✐st❡♥❝❡

❉❙②♥t❤(s, P❛ss❡❞) =                                   

r❡t✉r♥ ♥♦t❤✐♥❣ ✐❢ st❛t❡ ♠❡t ❜❡❢♦r❡

• ⊥ ✐❢ s ∈ P❛ss❡❞

♦t❤❡r✇✐s❡✿

r❡❝✉rs✐✈❡ ❝❛❧❧ ♦♥ t❤❡ s✉❝❝❡ss♦rs

• s′∈❙✉❝❝(s)

❉❙②♥t❤(s′, P❛ss❡❞ ∪ {s})

• ∪
• sC \
• s′∈❙✉❝❝(s)

❝❛♥ ♣❛ss t❤❡ ❣✉❛r❞

• sC ∧ g(s, s′)ւ

∧s′

C↓P)

• ↓P
• ✈❛❧✉❛t✐♦♥s ❢♦r ✇❤✐❝❤ ♦♥❡ ♠❛② ❣❡t st✉❝❦ ✐♥ s

s s′ s′′

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✹ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ ❆❧❣♦r✐t❤♠

❆ s❡♠✐✲❛❧❣♦r✐t❤♠ ❢♦r ❞❡❛❞❧♦❝❦✲❢r❡❡♥❡ss

P❉❋❈(A) = ¬❉❙②♥t❤(sA

0 , ∅)

❙❡♠✐✲❛❧❣♦r✐t❤♠✿ r❡s✉❧t ❝♦rr❡❝t ✇❤❡♥❡✈❡r ✐t t❡r♠✐♥❛t❡s ✭✇❤✐❝❤ ✐s ♥♦t ❣✉❛r❛♥t❡❡❞ ✕ ✉♥❞❡❝✐❞❛❜❧❡ ♣r♦❜❧❡♠✮

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✺ ✴ ✷✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣ ❆❧❣♦r✐t❤♠

❆ s❡♠✐✲❛❧❣♦r✐t❤♠ ❢♦r ❞❡❛❞❧♦❝❦✲❢r❡❡♥❡ss

P❉❋❈(A) = ¬❉❙②♥t❤(sA

0 , ∅)

❙❡♠✐✲❛❧❣♦r✐t❤♠✿ r❡s✉❧t ❝♦rr❡❝t ✇❤❡♥❡✈❡r ✐t t❡r♠✐♥❛t❡s ✭✇❤✐❝❤ ✐s ♥♦t ❣✉❛r❛♥t❡❡❞ ✕ ✉♥❞❡❝✐❞❛❜❧❡ ♣r♦❜❧❡♠✮

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✺ ✴ ✷✷

❇❛❝❦✇❛r❞ ❯♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥

❖✉t❧✐♥❡

✶

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣

✸

❇❛❝❦✇❛r❞ ❯♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥

✹

❊①♣❡r✐♠❡♥ts

✺

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✻ ✴ ✷✷

❇❛❝❦✇❛r❞ ❯♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥

❆ ❝♦♥s❡r✈❛t✐✈❡ ✉♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥ ❇✇❯❙

■❞❡❛✿

✶ ●❡♥❡r❛t❡ ❛ ♣❛rt✐❛❧ st❛t❡ s♣❛❝❡ ✷ ❈♦♥s✐❞❡r ✉♥❦♥♦✇♥ st❛t❡s ❛s ✉♥s❛❢❡ ✭❞❡❛❞❧♦❝❦❡❞✮ ✸ ❘❡❝♦♠♣✉t❡ t❤❡ ❞❡❛❞❧♦❝❦✲❢r❡❡♥❡ss ❝♦♥str❛✐♥t ✐♥ ❛ ❜❛❝❦✇❛r❞ ♠❛♥♥❡r

✉♥t✐❧ ✜①♣♦✐♥t s0 s1 s2 s3 s4 × × ×

✉♥❡①♣❧♦r❡❞ s✉❝❝❡ss♦rs

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✼ ✴ ✷✷

❇❛❝❦✇❛r❞ ❯♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥

❆ ❝♦♥s❡r✈❛t✐✈❡ ✉♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥ ❇✇❯❙

■❞❡❛✿

✶ ●❡♥❡r❛t❡ ❛ ♣❛rt✐❛❧ st❛t❡ s♣❛❝❡ ✷ ❈♦♥s✐❞❡r ✉♥❦♥♦✇♥ st❛t❡s ❛s ✉♥s❛❢❡ ✭❞❡❛❞❧♦❝❦❡❞✮ ✸ ❘❡❝♦♠♣✉t❡ t❤❡ ❞❡❛❞❧♦❝❦✲❢r❡❡♥❡ss ❝♦♥str❛✐♥t ✐♥ ❛ ❜❛❝❦✇❛r❞ ♠❛♥♥❡r

✉♥t✐❧ ✜①♣♦✐♥t s0 s1 s2 s3 s4 × × ×

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✼ ✴ ✷✷

❇❛❝❦✇❛r❞ ❯♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥

❆ ❝♦♥s❡r✈❛t✐✈❡ ✉♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥ ❇✇❯❙

■❞❡❛✿

✶ ●❡♥❡r❛t❡ ❛ ♣❛rt✐❛❧ st❛t❡ s♣❛❝❡ ✷ ❈♦♥s✐❞❡r ✉♥❦♥♦✇♥ st❛t❡s ❛s ✉♥s❛❢❡ ✭❞❡❛❞❧♦❝❦❡❞✮ ✸ ❘❡❝♦♠♣✉t❡ t❤❡ ❞❡❛❞❧♦❝❦✲❢r❡❡♥❡ss ❝♦♥str❛✐♥t ✐♥ ❛ ❜❛❝❦✇❛r❞ ♠❛♥♥❡r

✉♥t✐❧ ✜①♣♦✐♥t s0 s1 s2 s3 s4 × × ×

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✼ ✴ ✷✷

❇❛❝❦✇❛r❞ ❯♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥

❆ ❝♦♥s❡r✈❛t✐✈❡ ✉♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥ ❇✇❯❙

■❞❡❛✿

✶ ●❡♥❡r❛t❡ ❛ ♣❛rt✐❛❧ st❛t❡ s♣❛❝❡ ✷ ❈♦♥s✐❞❡r ✉♥❦♥♦✇♥ st❛t❡s ❛s ✉♥s❛❢❡ ✭❞❡❛❞❧♦❝❦❡❞✮ ✸ ❘❡❝♦♠♣✉t❡ t❤❡ ❞❡❛❞❧♦❝❦✲❢r❡❡♥❡ss ❝♦♥str❛✐♥t ✐♥ ❛ ❜❛❝❦✇❛r❞ ♠❛♥♥❡r

✉♥t✐❧ ✜①♣♦✐♥t s0 s1 s2 s3 s4 × × ×

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✼ ✴ ✷✷

❇❛❝❦✇❛r❞ ❯♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥

❆ ❝♦♥s❡r✈❛t✐✈❡ ✉♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥ ❇✇❯❙

■❞❡❛✿

✶ ●❡♥❡r❛t❡ ❛ ♣❛rt✐❛❧ st❛t❡ s♣❛❝❡ ✷ ❈♦♥s✐❞❡r ✉♥❦♥♦✇♥ st❛t❡s ❛s ✉♥s❛❢❡ ✭❞❡❛❞❧♦❝❦❡❞✮ ✸ ❘❡❝♦♠♣✉t❡ t❤❡ ❞❡❛❞❧♦❝❦✲❢r❡❡♥❡ss ❝♦♥str❛✐♥t ✐♥ ❛ ❜❛❝❦✇❛r❞ ♠❛♥♥❡r

✉♥t✐❧ ✜①♣♦✐♥t s0 s1 s2 s3 s4 × × ×

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✼ ✴ ✷✷

❊①♣❡r✐♠❡♥ts

❖✉t❧✐♥❡

✶

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣

✸

❇❛❝❦✇❛r❞ ❯♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥

✹

❊①♣❡r✐♠❡♥ts

✺

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✽ ✴ ✷✷

❊①♣❡r✐♠❡♥ts

■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥ ■▼■❚❆❚❖❘

❆ t♦♦❧ ❢♦r ♠♦❞❡❧✐♥❣ ❛♥❞ ✈❡r✐❢②✐♥❣ r❡❛❧✲t✐♠❡ s②st❡♠s ✇✐t❤ ✉♥❦♥♦✇♥ ❝♦♥st❛♥ts ♠♦❞❡❧❡❞ ✇✐t❤ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ❛✉t♦♠❛t❛

❈♦♠♠✉♥✐❝❛t✐♦♥ t❤r♦✉❣❤ ✭str♦♥❣✮ ❜r♦❛❞❝❛st s②♥❝❤r♦♥✐③❛t✐♦♥ ■♥t❡❣❡r✲✈❛❧✉❡❞ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ❙t♦♣✇❛t❝❤❡s✱ t♦ ♠♦❞❡❧ s❝❤❡❞✉❧❛❜✐❧✐t② ♣r♦❜❧❡♠s ✇✐t❤ ♣r❡❡♠♣t✐♦♥

❯♥❞❡r ❝♦♥t✐♥✉♦✉s ❞❡✈❡❧♦♣♠❡♥t s✐♥❝❡ ✷✵✵✽

❬❆♥❞ré ❡t ❛❧✳✱ ✷✵✶✷❪

❋r❡❡ ❛♥❞ ♦♣❡♥ s♦✉r❝❡ s♦❢t✇❛r❡✿ ❆✈❛✐❧❛❜❧❡ ✉♥❞❡r t❤❡ ●◆❯✲●P▲ ❧✐❝❡♥s❡

✇✇✇✳✐♠✐t❛t♦r✳❢r

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✶✾ ✴ ✷✷

❊①♣❡r✐♠❡♥ts

❙✉♠♠❛r② ♦❢ t❤❡ ❡①♣❡r✐♠❡♥ts

❈❛s❡ st✉❞② |A| |X| |P| ❙t❛t❡s P❉❋❈ ❇✇❯❙ K ❙♦✉♥❞♥❡ss ❋✐❣✳ ✶❛ ✶ ✶ ✷ ✸ 0.012 ✲ ♥♥❝❝ ❡①❛❝t ❋✐❣✳ ✶❜ ✶ ✶ ✶ ✷ 0.005 ✲ ⊥ ❡①❛❝t ❛♥❞✕♦r ❝✐r❝✉✐t ✹ ✹ ✹ ✺✱✷✻✺ ❚❖ 171 [♥♥❝❝−, ♥♥❝❝+] ✉♥❞❡r✴♦✈❡r✲❛♣♣ ❝♦✛❡❡ ♠❛❝❤✐♥❡ ✶ ✶ ✷ ✸ ✾✱✵✹✷ ❚❖ 8.4 [♥♥❝❝−, ♥♥❝❝+] ✉♥❞❡r✴♦✈❡r✲❛♣♣ ❝♦✛❡❡ ♠❛❝❤✐♥❡ ✷ ✷ ✸ ✸ ✺✶ 0.198 ✲ ♥♥❝❝ ❡①❛❝t ❈❙▼❆✴❈❉ ♣r♦t♦❝♦❧ ✸ ✸ ✸ ✸✽ 0.105 ✲ ⊥ ❡①❛❝t ✢✐♣✲✢♦♣ ❝✐r❝✉✐t ✻ ✺ ✷ ✷✵ 0.093 ✲ ⊥ ❡①❛❝t ♥✉❝❧❡❛r ♣❧❛♥t ✶ ✷ ✹ ✶✸ 0.014 ✲ ♥♥❝❝ ❡①❛❝t ❘❈P ♣r♦t♦❝♦❧ ✺ ✻ ✺ ✷✱✵✾✶ 10.63 ✲ ⊥ ❡①❛❝t ❙■▼❖P ✺ ✽ ✷ ✷✷✱✽✾✹ ❚❖ 121 ♥♥❝❝ ♦✈❡r✲❛♣♣ ❚r❛✐♥ ❝♦♥tr♦❧❧❡r ✶ ✷ ✸ ✶✶ 0.025 ✲ ♥♥❝❝ ❡①❛❝t ❲❋❆❙ ✸ ✹ ✷ ✶✹✱✻✶✹ ❚❖ 69.1 [♥♥❝❝−, ♥♥❝❝+] ✉♥❞❡r✴♦✈❡r✲❛♣♣

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✷✵ ✴ ✷✷

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

❖✉t❧✐♥❡

✶

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

✷

❉❡❛❞❧♦❝❦✲❈❤❡❝❦✐♥❣

✸

❇❛❝❦✇❛r❞ ❯♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥

✹

❊①♣❡r✐♠❡♥ts

✺

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✷✶ ✴ ✷✷

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

❈♦♥❝❧✉s✐♦♥ ❙❡♠✐✲❛❧❣♦r✐t❤♠ t♦ ❣✉❛r❛♥t❡❡ t❤❡ ❛❜s❡♥❝❡ ♦❢ ❞❡❛❞❧♦❝❦s ✐♥ t✐♠❡❞ ❛✉t♦♠❛t❛ ✇✐t❤ ♣❛r❛♠❡tr✐❝ t✐♠✐♥❣ ❝♦♥st❛♥ts ❯♥❞❡r✲❛♣♣r♦①✐♠❛t✐♦♥ ✇✐t❤ ❣✉❛r❛♥t❡❡❞ t❡r♠✐♥❛t✐♦♥ P❡rs♣❡❝t✐✈❡s ❚❤❡♦r②✿ ❞❡❝✐❞❛❜❧❡ s✉❜❝❧❛ss❡s ❢♦r ❞❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣❄ Pr❛❝t✐❝❡✿ ❡✣❝✐❡♥t ❝♦♠♣✉t❛t✐♦♥ ✭♠✉❧t✐✲❝♦r❡✮ ❘❡✉s❛❜✐❧✐t②✿ ❆♣♣❧② t❤❡ ✐❞❡❛ ❜❡❤✐♥❞ ❇✇❯❙ t♦ ♦t❤❡r ❛❧❣♦r✐t❤♠s

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✷✷ ✴ ✷✷

❇✐❜❧✐♦❣r❛♣❤②

❇✐❜❧✐♦❣r❛♣❤②

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✷✸ ✴ ✷✷

❇✐❜❧✐♦❣r❛♣❤②

❘❡❢❡r❡♥❝❡s ■

❆❧✉r✱ ❘✳ ❛♥❞ ❉✐❧❧✱ ❉✳ ▲✳ ✭✶✾✾✹✮✳ ❆ t❤❡♦r② ♦❢ t✐♠❡❞ ❛✉t♦♠❛t❛✳ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✶✷✻✭✷✮✿✶✽✸✕✷✸✺✳ ❆❧✉r✱ ❘✳✱ ❍❡♥③✐♥❣❡r✱ ❚✳ ❆✳✱ ❛♥❞ ❱❛r❞✐✱ ▼✳ ❨✳ ✭✶✾✾✸✮✳ P❛r❛♠❡tr✐❝ r❡❛❧✲t✐♠❡ r❡❛s♦♥✐♥❣✳ ■♥ ❙❚❖❈✱ ♣❛❣❡s ✺✾✷✕✻✵✶✳ ❆❈▼✳ ❆♥❞ré✱ ➱✳✱ ❈❤❛t❛✐♥✱ ❚✳✱ ❊♥❝r❡♥❛③✱ ❊✳✱ ❛♥❞ ❋r✐❜♦✉r❣✱ ▲✳ ✭✷✵✵✾✮✳ ❆♥ ✐♥✈❡rs❡ ♠❡t❤♦❞ ❢♦r ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ❛✉t♦♠❛t❛✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❋♦✉♥❞❛t✐♦♥s ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✷✵✭✺✮✿✽✶✾✕✽✸✻✳ ❆♥❞ré✱ ➱✳✱ ❋r✐❜♦✉r❣✱ ▲✳✱ ❑ü❤♥❡✱ ❯✳✱ ❛♥❞ ❙♦✉❧❛t✱ ❘✳ ✭✷✵✶✷✮✳ ■▼■❚❆❚❖❘ ✷✳✺✿ ❆ t♦♦❧ ❢♦r ❛♥❛❧②③✐♥❣ r♦❜✉st♥❡ss ✐♥ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✳ ■♥ ❋▼✱ ✈♦❧✉♠❡ ✼✹✸✻ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✸✸✕✸✻✳ ❙♣r✐♥❣❡r✳ ❆♥❞ré✱ ➱✳ ❛♥❞ ▲✐♠❡✱ ❉✳ ✭✷✵✶✻✮✳ ▲✐✈❡♥❡ss ✐♥ ▲✴❯✲♣❛r❛♠❡tr✐❝ t✐♠❡❞ ❛✉t♦♠❛t❛✳ ❙✉❜♠✐tt❡❞✳ ❍❡♥③✐♥❣❡r✱ ❚✳ ❆✳✱ ◆✐❝♦❧❧✐♥✱ ❳✳✱ ❙✐❢❛❦✐s✱ ❏✳✱ ❛♥❞ ❨♦✈✐♥❡✱ ❙✳ ✭✶✾✾✹✮✳ ❙②♠❜♦❧✐❝ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ❢♦r r❡❛❧✲t✐♠❡ s②st❡♠s✳ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ✶✶✶✭✷✮✿✶✾✸✕✷✹✹✳

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✷✹ ✴ ✷✷

❇✐❜❧✐♦❣r❛♣❤②

❘❡❢❡r❡♥❝❡s ■■

❍✉♥❡✱ ❚✳✱ ❘♦♠✐❥♥✱ ❏✳✱ ❙t♦❡❧✐♥❣❛✱ ▼✳✱ ❛♥❞ ❱❛❛♥❞r❛❣❡r✱ ❋✳ ❲✳ ✭✷✵✵✷✮✳ ▲✐♥❡❛r ♣❛r❛♠❡tr✐❝ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ♦❢ t✐♠❡❞ ❛✉t♦♠❛t❛✳ ❏♦✉r♥❛❧ ♦❢ ▲♦❣✐❝ ❛♥❞ ❆❧❣❡❜r❛✐❝ Pr♦❣r❛♠♠✐♥❣✱ ✺✷✲✺✸✿✶✽✸✕✷✷✵✳ ❏♦✈❛♥♦✈✐➣✱ ❆✳✱ ▲✐♠❡✱ ❉✳✱ ❛♥❞ ❘♦✉①✱ ❖✳ ❍✳ ✭✷✵✶✺✮✳ ■♥t❡❣❡r ♣❛r❛♠❡t❡r s②♥t❤❡s✐s ❢♦r t✐♠❡❞ ❛✉t♦♠❛t❛✳ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❙♦❢t✇❛r❡ ❊♥❣✐♥❡❡r✐♥❣✱ ✹✶✭✺✮✿✹✹✺✕✹✻✶✳

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✷✺ ✴ ✷✷

▲✐❝❡♥s✐♥❣

▲✐❝❡♥s✐♥❣

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✷✻ ✴ ✷✷

▲✐❝❡♥s✐♥❣

❙♦✉r❝❡ ♦❢ t❤❡ ❣r❛♣❤✐❝s ✉s❡❞ ■

❚✐t❧❡✿ ❙♠✐❧❡② ❣r❡❡♥ ❛❧✐❡♥ ❜✐❣ ❡②❡s ✭❛❛❛❤✮ ❆✉t❤♦r✿ ▲❛❞②♦❢❍❛ts ❙♦✉r❝❡✿ ❤tt♣s✿✴✴❝♦♠♠♦♥s✳✇✐❦✐♠❡❞✐❛✳♦r❣✴✇✐❦✐✴❋✐❧❡✿❙♠✐❧❡②❴❣r❡❡♥❴❛❧✐❡♥❴❜✐❣❴❡②❡s✳s✈❣ ▲✐❝❡♥s❡✿ ♣✉❜❧✐❝ ❞♦♠❛✐♥ ❚✐t❧❡✿ ❙♠✐❧❡② ❣r❡❡♥ ❛❧✐❡♥ ❜✐❣ ❡②❡s ✭❝r②✮ ❆✉t❤♦r✿ ▲❛❞②♦❢❍❛ts ❙♦✉r❝❡✿ ❤tt♣s✿✴✴❝♦♠♠♦♥s✳✇✐❦✐♠❡❞✐❛✳♦r❣✴✇✐❦✐✴❋✐❧❡✿❙♠✐❧❡②❴❣r❡❡♥❴❛❧✐❡♥❴❜✐❣❴❡②❡s✳s✈❣ ▲✐❝❡♥s❡✿ ♣✉❜❧✐❝ ❞♦♠❛✐♥

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✷✼ ✴ ✷✷

▲✐❝❡♥s✐♥❣

▲✐❝❡♥s❡ ♦❢ t❤✐s ❞♦❝✉♠❡♥t

❚❤✐s ♣r❡s❡♥t❛t✐♦♥ ❝❛♥ ❜❡ ♣✉❜❧✐s❤❡❞✱ r❡✉s❡❞ ❛♥❞ ♠♦❞✐✜❡❞ ✉♥❞❡r t❤❡ t❡r♠s ♦❢ t❤❡ ❧✐❝❡♥s❡ ❈r❡❛t✐✈❡ ❈♦♠♠♦♥s ❆ttr✐❜✉t✐♦♥✲❙❤❛r❡❆❧✐❦❡ ✹✳✵ ❯♥♣♦rt❡❞ ✭❈❈ ❇❨✲❙❆ ✹✳✵✮

✭▲

❆

❚ ❊❳ s♦✉r❝❡ ❛✈❛✐❧❛❜❧❡ ♦♥ ❞❡♠❛♥❞✮

❆✉t❤♦r✿ ➱t✐❡♥♥❡ ❆♥❞ré ❤tt♣s✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✲s❛✴✹✳✵✴

➱✳ ❆♥❞ré ✭❯✳ P❛r✐s ✶✸✮ ❉❡❛❞❧♦❝❦✲❝❤❡❝❦✐♥❣ P❚❆s ✷✻t❤ ❖❝t♦❜❡r ✷✵✶✻ ✷✽ ✴ ✷✷